Fermions on replica geometries and the $\Theta$-$\theta$ relation

Sunil Mukhi; Sameer Murthy

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arXiv · 1805.11114

Fermions on replica geometries and the $\Theta$-$\theta$ relation

Sunil Mukhi·Sameer Murthy

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Abstract

In arXiv:1706:09426 we conjectured and provided evidence for an identity between Siegel $\Theta$-constants for special Riemann surfaces of genus $n$ and products of Jacobi $\theta$-functions. This arises by comparing two different ways of computing the \nth \Renyi entropy of free fermions at finite temperature. Here we show that for $n=2$ the identity is a consequence of an old result due to Fay for doubly branched Riemann surfaces. For $n>2$ we provide a detailed matching of certain zeros on both sides of the identity. This amounts to an elementary proof of the identity for $n=2$, while for $n\ge 3$ it gives new evidence for it. We explain why the existence of additional zeros renders the general proof difficult.

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Sunil Mukhi, Sameer Murthy. 2018-05-28. Fermions on replica geometries and the $\Theta$-$\theta$ relation. https://arxiv.org/abs/1805.11114

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